Find the sum of the following A.P.'s :
(i) `3,8,13, .... ` to 20 terms. (ii) `1,4,7,.........` to 50 terms.
(iii) `8,5,2, ....` to 25 terms. (iv) `(a+b), (2a+3b), (3b+5b), ........ ` to n terms.

Answer

Let's solve the given problems step by step.

### (i) Find the sum of the A.P. `3, 8, 13, ...` to 20 terms.

1. **Identify the first term (A)**:
- The first term \( A = 3 \).

2. **Determine the common difference (D)**:
- The common difference \( D = 8 - 3 = 5 \).

3. **Identify the number of terms (N)**:
- The number of terms \( N = 20 \).

4. **Use the formula for the sum of the first N terms of an A.P.**:
\[ S_N = \frac{N}{2} \times (2A + (N-1)D) \]

5. **Substitute the values into the formula**:
\[ S_{20} = \frac{20}{2} \times (2 \times 3 + (20-1) \times 5) \]
\[ = 10 \times (6 + 19 \times 5) \]
\[ = 10 \times (6 + 95) \]
\[ = 10 \times 101 = 1010 \]

### (ii) Find the sum of the A.P. `1, 4, 7, ...` to 50 terms.

1. **Identify the first term (A)**:
- The first term \( A = 1 \).

2. **Determine the common difference (D)**:
- The common difference \( D = 4 - 1 = 3 \).

3. **Identify the number of terms (N)**:
- The number of terms \( N = 50 \).

4. **Use the sum formula**:
\[ S_N = \frac{N}{2} \times (2A + (N-1)D) \]

5. **Substitute the values**:
\[ S_{50} = \frac{50}{2} \times (2 \times 1 + (50-1) \times 3) \]
\[ = 25 \times (2 + 49 \times 3) \]
\[ = 25 \times (2 + 147) \]
\[ = 25 \times 149 = 3725 \]

### (iii) Find the sum of the A.P. `8, 5, 2, ...` to 25 terms.

1. **Identify the first term (A)**:
- The first term \( A = 8 \).

2. **Determine the common difference (D)**:
- The common difference \( D = 5 - 8 = -3 \).

3. **Identify the number of terms (N)**:
- The number of terms \( N = 25 \).

4. **Use the sum formula**:
\[ S_N = \frac{N}{2} \times (2A + (N-1)D) \]

5. **Substitute the values**:
\[ S_{25} = \frac{25}{2} \times (2 \times 8 + (25-1) \times (-3)) \]
\[ = \frac{25}{2} \times (16 - 72) \]
\[ = \frac{25}{2} \times (-56) \]
\[ = 25 \times (-28) = -700 \]

### (iv) Find the sum of the A.P. `(a+b), (2a+3b), (3a+5b), ...` to n terms.

1. **Identify the first term (A)**:
- The first term \( A = a + b \).

2. **Determine the common difference (D)**:
- The second term is \( 2a + 3b \) and the first term is \( a + b \).
- Thus, \( D = (2a + 3b) - (a + b) = a + 2b \).

3. **Identify the number of terms (N)**:
- The number of terms \( N = n \).

4. **Use the sum formula**:
\[ S_N = \frac{N}{2} \times (2A + (N-1)D) \]

5. **Substitute the values**:
\[ S_n = \frac{n}{2} \times (2(a + b) + (n-1)(a + 2b)) \]
\[ = \frac{n}{2} \times (2a + 2b + (n-1)(a + 2b)) \]
\[ = \frac{n}{2} \times (2a + 2b + (n-1)a + 2(n-1)b) \]
\[ = \frac{n}{2} \times ((2 + n - 1)a + (2 + 2(n - 1))b) \]
\[ = \frac{n}{2} \times (na + 2nb) \]

### Final Answers:
1. \( S_{20} = 1010 \)
2. \( S_{50} = 3725 \)
3. \( S_{25} = -700 \)
4. \( S_n = \frac{n}{2} \times (na + 2nb) \)